Are the price of calls and puts related?

Today,
we are going to explore an important relationship between the price of
call and put options that is often times misunderstood, or just plain
ignored by retail option traders. Understanding this concept is
essential to all levels of option traders.

Have
you ever noticed call and put options with the same strike price and
expiration date very rarely trade at the same price? The call is
sometimes more than the put or sometimes the put is more than the call.
There is a reason why this occurs but it is often misinterpreted by
novice investors as a bullish or bearish sign for the underlying stock.
It has nothing to do with the potential market direction of the stock,
and everything to do with synthetic relationships and carry costs of a
position. To start on the right foot let’s define these two terms.

Definition: Syn·thet·ic (sn-thtk)
- A financial instrument that is created artificially by simulating the
risk and rewards of that instrument with the combined features of a
collection of other assets.

Definition: Carry Costs (kr
kôsts) - Costs incurred as a result of an investment position. These
costs can include financial costs, interest on loans used to purchase a
the investment, and dividends that are expected to paid by some
securities.

The
core concept of why we will be using these terms today is that some
option strategies have very similar risks and rewards to an entirely
different strategy except for the carry costs of the position. Bottom
line, if you include carry costs when pricing option contracts you can
find two totally different strategies that are truly synthetic versions
of each other. To prove this let’s ponder this question.

Can X actually equal Y?

Sometimes,
yes. Consider the risk, rewards and market outlook of the following
pair: Long Stock plus a Long Put. In many ways, this resembles holding
just a long call: you benefit if the market rises, and you have limited
and known risk if the market falls.

The
profit and loss graphs below make this similarity a little clearer. The
X-axis refers to the stock price at expiration, and the y-axis refers
to the profit or loss for each position. Let's assume the stock is at
80, the interest rate is 5.3613%, volatility is 27.95% and expiration is
61 days away, with no dividend upcoming. Let's also assume we're using
the ATM put and call in this example, with a strike price for each of
80. (Note: The interest rate is large for the current market
environment, but it helps with the understanding of the concept.)

The
first graph shows the P&L of a long stock position (gray), the
P&L for a long put (green), and the P&L for the combined
position (red). Compare this red line with the second graph, which
depicts the simple P&L of the long call (blue). It's easy to see
that buying the protective put and buying the call results in a very
similar P&L at expiration.

Before
we run away and say they're not just similar but identical, let's graph
both on the same graph and see if they are any subtle differences.

When
we graph each position on top of the other (protective put in red and
long call in blue), the point where each strategy crosses the X-axis is
called the breakeven point at expiration. Notice the small difference in
the breakeven number of these two positions (83.35 vs 84). That's
because the put is trading for a little less than the call. Why? It has
to or there would be an arbitrage situation. Let me explain: we have two
strategies that are mostly the same as far as risk and reward are
concerned, but not with regard to carry costs. Quick interruption for another question:

Using
a similar example to the one above, lets imagine you had enough money
to place either trade. You could either buy the protective put or just
buy the call outright. But in this example let’s say the 80 strike put
and 80 strike call were trading for the same amount, which trade would
you use your funds on: a protective put or a long call? I hope you said
long call, because if the 80 strike put and 80 strike call are exactly
the same price, it makes sense to choose the call. With the call, you
won't have to spend additional money to buy the stock (remember, a
protective put is a long put PLUS long stock), and because the put and
call are the same price the breakevens would be exactly the same number
for either strategy (as opposed to the positions portrayed in the graphs
above).

Back to the original example:

As
noted in the graphs above the breakevens are not exactly the same, so
why? Since both trades have basically the same risk and reward, to make
them truly comparable we have to even things out. How? If we take the
carry-cost of the stock out of the price of the put, then everything
comes into equal balance. Here is the quick and dirty math involved:

If
we didn’t have to buy the stock we could put this 8000 dollars into an
interest-bearing account at 5.3613% (for this example) and we would earn
$428.90 annually in interest. By holding long stock, we're forgoing
that interest income: that's what "cost-to-carry" actually means.

Translate to days = 428.90/365 = 1.17 of interest income lost per dayCarry-costs over life of the option positions = 1.17 x 61 days to expiration = 71.37/100 = 0.71 or 71 cents

Compare
this number, 0.71 to the different between breakevens in our graph
above, which is 0.75 (84 - 83.35 = 0.75). That's pretty much identical
if you account for rounding errors. Even though 75 cents might seem like
a small discrepancy, it's an important one. In fact, this relationship
has to hold true or institutional traders would buy the cheap strategy
and sell the expensive strategy and collect money above and beyond the
carry-cost with no additional risk. That's the arbitrage opportunity I
referred to before.Time to play market-maker:

To
understand why this matters, put on your mesh jacket and imagine you're
a market-maker for a second. An apparently tiny difference like this
matters more to market makers than retail customers because of their
role to provide liquidity to the marketplace. That role often boxes them
into positions they didn't plan on holding, so they're always
considering alternative "ways out" of various trades. This is why
knowing how to create a position synthetically is a must for market
makers.

Market
makers are also looking for the smallest edge they can possibly get.
Given their huge size and very low transaction costs, it can make
financial sense for them to find a trade that makes them a penny and try
to make that penny 10,000 times in a single day. This explains why
calls and puts on the same strike line and with the same expiration
don’t usually trade for the same price.

Now let us toss a stock dividend into the mix:

So
what if a stock above pays a dividend before the expiration of the
option contacts? Better yet, what if it is a big dividend - like $1.00
per share? A dollar per share paid dividend paid by the stock is more than the interest
cost that could be earned by keeping the $8,000 in an interest-bearing
account over the life of the option contracts. We already determined the interest earned on the cash
to be about $71, a dollar dividend on 100 shares of stock would net
$100.

If
this is the case, it becomes more beneficial to own the stock outright
than have the cash. This implies it is cheaper
overall to carry the long stock and long put position than it is to carry the
long call position. Therefore, the put has to trade for more than the
call in this scenario to “balance” things out.

If
you would like to run your own examples inside your TradeKing account,
just use the options calculator under the Tools menu (after you are
signed in) and put a stock in that pays a dividend. Remove the dividend,
maybe change the interest rate and hit “re-calculate” and note what
happens to the theoretical price of the call and put on the right hand
side.

Tacked another misunderstanding in the option marketplace:

That's it for today, folks. Next we'll explore other synthetic relationships that exist in the complex world of option trading.

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While
Delta represents the consensus of the marketplace as to the theoretical
price movement of the option relative to the underlying security there
is no guarantee that this forecast will be correct.

While
Gamma represents the consensus of the marketplace as to the theoretical
rate of change of Delta relative to the underlying security there is no
guarantee that this forecast will be correct.

While
Theta represents the consensus of the marketplace as to the amount a
theoretical option's price will change for a corresponding one-unit
(day) change in the days to expiration of the option contract there is
no guarantee that this forecast will be correct.

While
Alpha represents the consensus of the marketplace as to the
relationship between gamma and theta there is no guarantee that this
forecast will be correct.

While
Vega represents the consensus of the marketplace as to the amount a
theoretical option's price will change for a corresponding one-unit
(point) change the implied volatility of the option contract there is no
guarantee that this forecast will be correct.

While
Rho represents the consensus of the marketplace as to the amount a
theoretical option's price will change for a corresponding one-unit
(percent) change in the interest rate used to price the option contract
there is no guarantee that this forecast will be correct.

While
implied volatility represents the consensus of the marketplace as to
the future level of stock price volatility or probability of reaching a
specific price point there is no guarantee that this forecast will be
correct.

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